Question

Решите неравенство $| frac{x-5}{3x^2-5x-2}| geq 1$

Answer 1

$mathtt{|frac{x-5}{3x^2-5x-2}|geq1~toleft[begin{array}{ccc}mathtt{frac{x-5}{3x^2-5x-2}+1leq0}\mathtt{frac{x-5}{3x^2-5x-2}-1geq0}end{array}rightto~left[begin{array}{ccc}mathtt{frac{x-5+(3x^2-5x-2)}{3x^2-5x-2}leq0}\mathtt{frac{x-5-(3x^2-5x-2)}{3x^2-5x-2}geq0}end{array}rightto~\}$$\mathtt{left[begin{array}{ccc}mathtt{frac{3x^2-4x-7}{3x^2-5x-2}leq0}\mathtt{frac{3x^2-6x+3}{3x^2-5x-2}leq0}end{array}rightto~left[begin{array}{ccc}mathtt{frac{(x+1)(x-frac{7}{3})}{(x+frac{1}{3})(x-2)}leq0}\mathtt{frac{(x-1)^2}{(x+frac{1}{3})(x-2)}leq0}end{array}rightto~left[begin{array}{ccc}mathtt{xin[-1;-frac{1}{3})U(2;frac{7}{3}]}\mathtt{xin(-frac{1}{3};2)}end{array}right}$

ответ: $mathtt{xin[-1;-frac{1}{3})U(-frac{1}{3};2)U(2;frac{7}{3}]}$